Topological phases of parafermions: a model with exactly-solvable ground states

Parafermions are the simplest generalization of Majorana fermions: they show non-Abelian fractional statistics and can realize topological order. We present a non-trivial and quasi-exactly-solvable model for a chain of parafermions in a symmetry-protected topological phase. We characterize analytically the ground-state wavefunctions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wavefunctions, we demonstrate analytically several signatures of topological order such as non-local edge-edge correlations, the presence of edge zero modes and the threefold degeneracy of the entanglement spectrum. Our model also provides a typical example of weak edge modes which are not commuting with the full Hamiltonian.